Life

Klein during his Leipzig period.

Felix Klein was born on 25 April 1849 in Düsseldorf,[1] to Prussian parents; his father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. Klein's mother was Sophie Elise Klein (1819–1890, née Kayser).[2] He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn,[3] 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, during 1866, Plücker's interest was geometry. Klein received his doctorate, supervised by Plücker, from the University of Bonn during 1868.

Plücker died during 1868, leaving his book concerning the basis of line geometry incomplete. Klein was the obvious person to complete the second part of Plücker's Neue Geometrie des Raumes, and thus became acquainted with Alfred Clebsch, who had relocated to Göttingen during 1868. Klein visited Clebsch the next year, along with visits to Berlin and Paris. During July 1870, at the beginning of the Franco-Prussian War, he was in Paris and had to leave the country. For a brief time he served as a medical orderly in the Prussian army before being appointed lecturer at Göttingen during early 1871.

After five years at the Technische Hochschule, Klein was appointed to a chair of geometry at Leipzig. There his colleagues included Walther von Dyck, Rohn, Eduard Study and Friedrich Engel. Klein's years at Leipzig, 1880 to 1886, fundamentally changed his life. During 1882, his health collapsed; during 1883–1884, he was plagued by depression. Nonetheless his research continued; his seminal work on hyperelliptic sigma functions dates from around this period, being published during 1886 and 1888.

Klein accepted a professorship at the University of Göttingen during 1886. From then until his 1913 retirement, he sought to re-establish Göttingen as the world's main mathematics research center. Yet he never managed to transfer from Leipzig to Göttingen his own primacy as a developer of geometry. At Göttingen, he taught a variety of courses, mainly concerning the interface between mathematics and physics, such as mechanics and potential theory.

The research facility Klein established at Göttingen served as a model for the best such facilities throughout the world. He introduced weekly discussion meetings, and created a mathematical reading room and library. During 1895, Klein hired David Hilbert away from the University of Königsberg; this appointment proved fateful, because Hilbert continued Göttingen's good reputation until his own retirement during 1932.

Work

Klein's first important mathematical discoveries were made during 1870. In collaboration with Sophus Lie, he discovered the fundamental properties of the asymptotic lines on the Kummer surface. They later investigated W-curves, curves invariant under a group of projective transformations. It was Lie who introduced Klein to the concept of group, which was to have a major role in his later work. Klein also learned about groups from Camille Jordan.[11]

A hand-blown Klein Bottle

Klein devised the "Klein bottle" named after him, a one-sided closed surface which cannot be embedded in three-dimensional Euclidean space, but it may be immersed as a cylinder looped back through itself to join with its other end from the "inside". It may be embedded in the Euclidean space of dimensions 4 and higher. The concept of a Klein Bottle was devised as a 3-Dimensional Möbius strip, with one method of construction being the attachment of the edges of two Möbius strips.[12]

Erlangen program

During 1871, while at Göttingen, Klein made major discoveries in geometry. He published two papers On the So-called Non-Euclidean Geometry showing that Euclidean and non-Euclidean geometries could be considered metric spaces determined by a Cayley-Klein metric. This insight had the corollary that non-Euclidean geometry was consistent if and only if Euclidean geometry was, giving the same status to geometries Euclidean and non-Euclidean, and ending all controversy about non-Euclidean geometry. Arthur Cayley never accepted Klein's argument, believing it to be circular.

Klein's synthesis of geometry as the study of the properties of a space that is invariant under a given group of transformations, known as the Erlangen Program (1872), profoundly influenced the evolution of mathematics. This program was initiated by Klein's inaugural lecture as professor at Erlangen, although it was not the actual speech he gave on the occasion. The program proposed a unified system of geometry that has become the accepted modern method. Klein showed how the essential properties of a given geometry could be represented by the group of transformations that preserve those properties. Thus the program's definition of geometry encompassed both Euclidean and non-Euclidean geometry.

Presently the significance of Klein's contributions to geometry is more than evident, but not because those contributions are now considered strange or wrong. On the contrary, those contributions have become so much a part of our present mathematical thinking that it is difficult for us to appreciate their novelty, and the way in which they were not immediately accepted by all his contemporaries.

Complex analysis

Klein saw his work on complex analysis as his major contribution to mathematics, specifically his work on:

Klein considered equations of degree > 4, and was especially interested in using transcendental methods to solve the general equation of the fifth degree. Building on the methods of Charles Hermite and Leopold Kronecker, he produced similar results to those of Brioschi and later completely solved the problem by means of the icosahedral group. This work enabled him to write a series of papers on elliptic modular functions.

In his 1884 book on the icosahedron, Klein established a theory of automorphic functions, associating algebra and geometry. However Poincaré published an outline of his theory of automorphic functions during 1881, which resulted in a friendly rivalry between the two men. Both sought to state and prove a grand uniformization theorem that would establish the new theory more completely. Klein succeeded in formulating such a theorem and in describing a strategy for proving it. But while doing this work his health decreased, as mentioned above.

This page is based on a Wikipedia article written by authors
(here).
Text is available under the CC BY-SA 3.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.